`Journal of Applied MathematicsVolume 2012, Article ID 348654, 9 pageshttp://dx.doi.org/10.1155/2012/348654`
Research Article

## A Two-Step Matrix-Free Secant Method for Solving Large-Scale Systems of Nonlinear Equations

1Department of Mathematics, Faculty of Science, University Putra Malaysia, 43400 Serdang, Malaysia
2Department of Mathematics, Faculty of Science, Bayero University Kano, Kano 2340, Nigeria
3Department of Mathematics, Faculty of Science and Technology, Malaysia Terengganu University, Kuala Lumpur 21030 Terengganu, Malaysia

Received 9 November 2011; Revised 11 January 2012; Accepted 16 January 2012

Copyright © 2012 M. Y. Waziri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. J. E. Dennis, Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, NJ, USA, 1983.
2. R. S. Dembo, S. C. Eisenstat, and T. Steihaug, “Inexact Newton methods,” SIAM Journal on Numerical Analysis, vol. 19, no. 2, pp. 400–408, 1982.
3. K. Natasa and L. Zorna, “Newton-like method with modification of the righ-thand vector,” Journal of Computational Mathematics, vol. 71, pp. 237–250, 2001.
4. M. Y. Waziri, W. J. Leong, M. A. Hassan, and M. Monsi, “A low memory solver for integral equations of chandrasekhar type in the radiative transfer problems,” Mathematical Problems in Engineering, vol. 2011, Article ID 467017, 12 pages, 2011.
5. B. Lam, “On the convergence of a quasi-Newton method for sparse nonlinear systems,” Mathematics of Computation, vol. 32, no. 142, pp. 447–451, 1978.
6. W. J. Leong, M. A. Hassan, and M. Waziri Yusuf, “A matrix-free quasi-Newton method for solving large-scale nonlinear systems,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2354–2363, 2011.
7. J. E. Dennis, Jr. and H. Wolkowicz, “Sizing and least-change secant methods,” SIAM Journal on Numerical Analysis, vol. 30, no. 5, pp. 1291–1314, 1993.
8. J. A. Ford and I. A. Moghrabi, “Alternating multi-step quasi-Newton methods for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 82, no. 1-2, pp. 105–116, 1997, 7th ICCAM 96 Congress (Leuven).
9. J. A. Ford and I. A. Moghrabi, “Multi-step quasi-Newton methods for optimization,” Journal of Computational and Applied Mathematics, vol. 50, no. 1–3, pp. 305–323, 1994.
10. M. Farid, W. J. Leong, and M. A. Hassan, “A new two-step gradient-type method for large-scale unconstrained optimization,” Computers & Mathematics with Applications, vol. 59, no. 10, pp. 3301–3307, 2010.
11. J. A. Ford and S. Thrmlikit, “New implicite updates in multi-step quasi-Newton methods for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 152, pp. 133–146, 2003.
12. B.-C. Shin, M. T. Darvishi, and C.-H. Kim, “A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3190–3198, 2010.
13. E. Spedicato, “Cumputational experience with quasi-Newton algorithms for minimization problems of moderatetly large size,” Tech. Rep. CISE-N-175 3, pp. 10–41, 1975.
14. A. Roose, V. L. M. Kulla, and T. Meressoo, Test Examples of Systems of Nonlinear Equations., Estonian Software and Computer Service Company, Tallin, Estonia, 1990.